diff --git a/src/Lean/Meta/Tactic/Grind/Arith/Linear/StructId.lean b/src/Lean/Meta/Tactic/Grind/Arith/Linear/StructId.lean
index 8814ffe080..9e4f8de57b 100644
--- a/src/Lean/Meta/Tactic/Grind/Arith/Linear/StructId.lean
+++ b/src/Lean/Meta/Tactic/Grind/Arith/Linear/StructId.lean
@@ -102,7 +102,8 @@ private def mkLawfulOrderLTInst? (u : Level) (type : Expr) (ltInst? leInst? : Op
   let some leInst := leInst? | return none
   let lawfulOrderLTType := mkApp3 (mkConst ``Std.LawfulOrderLT [u]) type ltInst leInst
   let some inst ← synthInstance? lawfulOrderLTType
-    | reportIssue! "type has `LE` and `LT`, but the `LT` instance is not lawful, failed to synthesize{indentExpr lawfulOrderLTType}"
+    | reportIssue! "type has `LE` and `LT`, but the `LT` instance is not lawful, failed to synthesize{indentExpr lawfulOrderLTType}\n\
+        the `linarith` module will not process strict inequalities in this type"
       return none
   return some inst
 
@@ -110,7 +111,8 @@ private def mkIsPreorderInst? (u : Level) (type : Expr) (leInst? : Option Expr)
   let some leInst := leInst? | return none
   let isPreorderType := mkApp2 (mkConst ``Std.IsPreorder [u]) type leInst
   let some inst ← synthInstance? isPreorderType
-    | reportIssue! "type has `LE`, but is not a preorder, failed to synthesize{indentExpr isPreorderType}"
+    | reportIssue! "type has `LE`, but is not a preorder, failed to synthesize{indentExpr isPreorderType}\n\
+        the `linarith` module will not run on this type"
       return none
   return some inst
 
@@ -118,7 +120,8 @@ private def mkIsPartialOrderInst? (u : Level) (type : Expr) (leInst? : Option Ex
   let some leInst := leInst? | return none
   let isPartialOrderType := mkApp2 (mkConst ``Std.IsPartialOrder [u]) type leInst
   let some inst ← synthInstance? isPartialOrderType
-    | reportIssue! "type has `LE`, but is not a partial order, failed to synthesize{indentExpr isPartialOrderType}"
+    | -- There is no need to report an issue here, as behaviour doesn't change: we only synthesize the partial order instance
+      -- in order to check the linear order and preorder instances are compatible.
       return none
   return some inst
 
@@ -126,7 +129,8 @@ private def mkIsLinearOrderInst? (u : Level) (type : Expr) (leInst?  : Option Ex
   let some leInst := leInst? | return none
   let isLinearOrderType := mkApp2 (mkConst ``Std.IsLinearOrder [u]) type leInst
   let some inst ← synthInstance? isLinearOrderType
-    | reportIssue! "type has `LE`, but is not a linear order, failed to synthesize{indentExpr isLinearOrderType}"
+    | reportIssue! "type has `LE`, but is not a linear order, failed to synthesize{indentExpr isLinearOrderType}\n\
+        the `linarith` module will not process disequalities in this type (or equality goals)"
       return none
   return some inst
 
@@ -137,7 +141,8 @@ private def mkOrderedRingInst? (u : Level) (type : Expr) (semiringInst? leInst?
   let some preorderInst := preorderInst? | return none
   let isOrdType := mkApp5 (mkConst ``Grind.OrderedRing [u]) type semiringInst leInst ltInst preorderInst
   let some inst ← synthInstance? isOrdType
-    | reportIssue! "type has a `Preorder` and is a `Semiring`, but is not an ordered ring, failed to synthesize{indentExpr isOrdType}"
+    | -- TODO: this issue message should explain which behaviours are disabled when then ordered ring instance is not available.
+      reportIssue! "type has a `Preorder` and is a `Semiring`, but is not an ordered ring, failed to synthesize{indentExpr isOrdType}"
       return none
   return some inst
 
@@ -288,7 +293,7 @@ where
     let id := (← get').structs.size
     let struct : Struct := {
       id, type, u, intModuleInst, leInst?, ltInst?, lawfulOrderLTInst?, isPreorderInst?,
-      orderedAddInst?, isPartialInst?, isLinearInst?, noNatDivInst?
+      orderedAddInst?, isLinearInst?, noNatDivInst?,
       leFn?, ltFn?, addFn, subFn, negFn, zsmulFn, nsmulFn, zsmulFn?, nsmulFn?, zero, one?
       ringInst?, commRingInst?, orderedRingInst?, charInst?, ringId?, fieldInst?, ofNatZero, homomulFn?
     }
diff --git a/src/Lean/Meta/Tactic/Grind/Arith/Linear/Types.lean b/src/Lean/Meta/Tactic/Grind/Arith/Linear/Types.lean
index 084c71bcf1..c901bab209 100644
--- a/src/Lean/Meta/Tactic/Grind/Arith/Linear/Types.lean
+++ b/src/Lean/Meta/Tactic/Grind/Arith/Linear/Types.lean
@@ -109,8 +109,6 @@ structure Struct where
   isPreorderInst?    : Option Expr
   /-- `OrderedAdd` instance with `IsPreorder` if available -/
   orderedAddInst?    : Option Expr
-  /-- `IsPartialOrder` instance if available -/
-  isPartialInst?     : Option Expr
   /-- `IsLinearOrder` instance if available -/
   isLinearInst?      : Option Expr
   /-- `NoNatZeroDivisors` -/
diff --git a/tests/lean/run/grind_linarith_1.lean b/tests/lean/run/grind_linarith_1.lean
index 248beb5e15..bac23ee932 100644
--- a/tests/lean/run/grind_linarith_1.lean
+++ b/tests/lean/run/grind_linarith_1.lean
@@ -68,10 +68,10 @@ example [CommRing α] [LE α] [LT α] [LawfulOrderLT α] [IsPreorder α] [Ordere
 /--
 trace: [grind.issues] type has `LE` and `LT`, but the `LT` instance is not lawful, failed to synthesize
       LawfulOrderLT α
-[grind.issues] type has `LE`, but is not a partial order, failed to synthesize
-      IsPartialOrder α
+    the `linarith` module will not process strict inequalities in this type
 [grind.issues] type has `LE`, but is not a linear order, failed to synthesize
       IsLinearOrder α
+    the `linarith` module will not process disequalities in this type (or equality goals)
 [grind.issues] type has a `Preorder` and is a `Semiring`, but is not an ordered ring, failed to synthesize
       OrderedRing α
 -/